Appendix S2. Model assessment
We conducted a thorough model assessment to understand the behavior and performance
of an integrated population model (IPM) for population dynamics of Wisconsin wolves. The
purpose of the model was to estimate a correction factor that acts as a discrepancy between the
estimates that the two data types would have provided if used independently. Here, we provide
an empirical justification of our IPM for estimating a correction factor, model checking with
external validation, and posterior predictive checking to establish whether our model is
consistent with our data. Finally, we conducted a sensitivity analysis to determine whether our
model was sensitive to our choice of prior distributions and model for the population count.
Empirical justification of an IPM for estimating a correction factor
We provide an empirical justification of the IPM estimating a correction factor as a
discrepancy between the estimates of the different data types (recruitment and survival data;
population count data). Our justification is based on whether the parameter estimates for
survival and recruitment from the IPM are the same as the parameter estimates of survival and
recruitment in an unintegrated model. When we run the survival model and the recruitment
model by themselves (and separate from each other) and not as part of the IPM we would expect
the parameter estimates to be the same as the parameter estimates from the IPM. This finding
would demonstrate that the correction factor that we are estimating is actually capturing that
discrepancy between the estimates of the different data types. If we do not find this
correspondence in the parameter estimates between the IPM and the unintegrated models, then
we would conclude that our correction factor was not acting as a discrepancy between the
estimates of the different data types.
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We ran the unintegrated model and compared the results for recruitment and survival to
the integrated model. First, we ran the recruitment portion of the model by itself (without the
survival information or population count information) and plotted the annual posterior means in
red in Figure S2.1. Next, we ran the unintegrated survival model by itself and plotted the
posterior mean in red with a dotted line in Figure S2.1. The posterior estimates from our
integrated model are essentially identical to the posterior estimates from our unintegrated model
(Figure S2.1). This provides the empirical justification for establishing our correction factor as a
discrepancy between the estimates. If these two data types were used separately, we would have
gotten the same estimates for recruitment and survival as we got in the integrated model.
Figure S2.1. Observed recruitment and posterior estimates and 95% credibility intervals for
annual recruitment, survival, and an estimated correction factor in the Wisconsin, USA wolf
population from 1980 – 2012 (Figure 3 from main text). The red line is the mean posterior
estimates of recruitment for each year when recruitment was run alone, and the red dotted line is
the mean posterior estimate of the survival rate when the survival data are run alone.
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External validation
We implemented an external validation for our IPM. The validation data that we left out
of the model was the population size in 2012. We predicted the 2012 population size from the
posterior distribution:
𝑁2012 ~ 𝑙𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙(𝜇2012 , 𝜐 2 )
𝜇2012 = log(𝑁2011 × (𝜌2012 + 𝜎2012 + 𝜅2012 ))
We concluded that if the mean of the posterior 𝑁2012 was contained within the estimated 2012
population size of 815 – 880 wolves, then our model provided reasonable inference.
The mean of the posterior 𝑁2012 = 878 which was within the estimated 2012 population
size of 815 – 880 wolves. Therefore, we concluded that the inferences from our model made
sense.
Posterior predictive checking
We used a graphical approach to the posterior predictive check and looked for
discrepancies between the real and simulated data (Gelman et al., 2003). We plotted the
exponential of the mean of the posteriors 𝜇𝑡 for 𝑡 = 2, 3, … 32 along with the observed mean
wolf population counts from Wisconsin from 1981 – 2011. We assessed whether the simulated
data matched the observed data and whether there were any systematic discrepancies in the
simulated data compared to the real data. If in the graphical assessment we saw close
correspondence between the simulated and real datasets, then we determined that the data fit the
model well. To look for normality in the residuals in the population count model, we plotted
(observed – expected) versus expected and looked for patterns in the residuals as evidence for
poor model fit. The observed data were the posterior means of 𝜇𝑡 for 𝑡 = 2, 3, … 32 and the
expected data were the log population counts from Wisconsin from 1981 – 2011.
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We plotted the exponential of the mean of the posteriors (red line, Fig. S2.2) along with
the observed population counts in 1981 – 2011 (open circles, Fig. S2.2) and found close
correspondence. Also, we plotted the residuals (the difference between the mean of the
posteriors and the log observed population plotted against the log observed population value) and
found that there was no pattern in the residuals (Fig. S2.3).
Figure S2.2. Wisconsin wolf population size from 1981 – 2011 as estimated from the means of
the posteriors in an IPM compared to what was observed.
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Figure S2.3. Residual plot (observed – expected versus expected) for population count data from
an IPM of wolf population dynamics from Wisconsin, USA.
Sensitivity analysis
We ran a sensitivity analysis on our choice of prior parameters in the survival model, the
distribution of population counts, and the prior distribution on the correction factor. In the
survival portion of the IPM, we chose a value of 𝑟 = 0.01, which corresponds to our prior guess
at the failure rate, and 𝑐 = 0.001, which corresponds to our confidence in our guess about the
failure rate (lower values of 𝑐 mean a lower confidence in our guess). In a one-factor-at-a-time
sensitivity analysis, we varied the values of 𝑐 = {0.1, 0.01,0.0001} and 𝑟 = {0.05, 0.2} in the
IPM and assessed whether the posterior estimates for survival were affected by our prior choice
of 𝑟 and 𝑐. To explore whether our choice of prior for the correction factor affected the posterior
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estimates, we ran the model with 𝜅 ~ 𝑢𝑛𝑖𝑓𝑜𝑟𝑚(0,1) which would assume that the correction
factor was always an addition to the population, and we ran the model with 𝜅 ~ 𝑢𝑛𝑖𝑓𝑜𝑟𝑚(−1,0)
to assume that the correction factor was always a subtraction from the population. Also, we ran
the IPM with a Poisson distribution for the population counts (instead of lognormal). In the
sensitivity analysis for the correction factor and population counts, we assessed whether our
choice of distribution led to a change in the posterior parameter estimates.
For the survival portion of the sensitivity analysis, we found that the mean and SD of the
posterior survival rate were robust to our choice of 𝑟 and 𝑐 (Table S2.1). When we used a
Poisson distribution for the population counts, we found that the mean parameter estimates were
very similar to the original model (lognormal distribution for population counts), except for an
increase in variance on the correction factors (Fig. S2.4). A change in the prior on the correction
factor to a Uniform(0,1) had the effect of reducing the mean survival rate and increasing the
mean correction factor compared to the original model (Fig. S2.5). A change in the prior on the
correction factor to a Uniform(-1,0) had the effect of increasing the mean survival rate and
decreasing the mean correction factor compared to the original model (Fig. S2.6). These results
were not surprising and we highlight our use of a Uniform(-1,1) for the correction factor in our
original model so that we are not making an unfounded assumption about the correction being a
net addition or subtraction from the population.
Table S2.1. Posterior mean and standard deviation (SD) of annual wolf survival rate from radiocollared wolves in Wisconsin, USA, under different prior values of r and c (parameters used in
the construction of the survival rate priors).
Mean of posterior
SD of posterior
Value of 𝑟
Value of 𝑐
0.1
0.001
0.7688
0.0141
0.1
0.0001
0.7688
0.0141
0.1
0.01
0.7686
0.0144
0.1
0.1
0.7652
0.0140
0.05
0.001
0.7688
0.0139
0.2
0.001
0.7686
0.0141
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Figure S2.4. Observed recruitment and posterior estimates and 95% credibility intervals for
annual recruitment, survival, and an estimated correction factor in the Wisconsin, USA wolf
population from 1980 – 2012 when population counts were modeled with a Poisson
distribution. The red lines (solid, dashed, and dotted) are the corresponding mean posterior
estimates from the original model (Figure 3 of main text).
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Figure S2.5. Observed recruitment and posterior estimates and 95% credibility intervals for
annual recruitment, survival, and an estimated correction factor in the Wisconsin, USA wolf
population from 1980 – 2012 when the prior on the correction factor was restricted to
Uniform(0,1). The red lines (solid, dashed, and dotted) are the corresponding mean posterior
estimates from the original model (Figure 3 of main text).
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Figure S2.6. Observed recruitment and posterior estimates and 95% credibility intervals for
annual recruitment, survival, and an estimated correction factor in the Wisconsin, USA wolf
population from 1980 – 2012 when the prior on the correction factor was restricted to Uniform(1,0). The red lines (solid, dashed, and dotted) are the corresponding mean posterior estimates
from the original model (Figure 3 of main text).
References
Gelman, A., Carlin, J.B., Stern, H.S. & Rubin, D.B. (2003) Bayesian data analysis. CRC press.
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